Simplify the following expression and state the condition under which the simplification is valid. $q = \dfrac{-p^2 - 3p + 54}{-10p^3 - 60p^2 + 270p}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ q = \dfrac {-1(p^2 + 3p - 54)} {-10p(p^2 + 6p - 27)} $ $ q = \dfrac{1}{10p} \cdot \dfrac{p^2 + 3p - 54}{p^2 + 6p - 27} $ Next factor the numerator and denominator. $ q = \dfrac{1}{10p} \cdot \dfrac{(p + 9)(p - 6)}{(p + 9)(p - 3)}$ Assuming $p \neq -9$ , we can cancel the $p + 9$ $ q = \dfrac{1}{10p} \cdot \dfrac{p - 6}{p - 3}$ Therefore: $ q = \dfrac{ p - 6 }{ 10p(p - 3)}$, $p \neq -9$